📚 IB AQA Mathematics: Mastering Integration | IB AQA 数学：积分考点精讲
Integration is a cornerstone of IB Mathematics, bridging the gap between accumulation and rate of change. Whether you are tackling the IB Analysis and Approaches course, understanding integration is essential for success in calculus, and it appears frequently in AQA-style exam questions. This guide breaks down every major integration technique, key formulas, and applications with bilingual explanations.
积分是 IB 数学的基石，连接着累积与变化率。无论你学习的是 IB 分析与方法课程，掌握积分都是微积分成功的关键，在 AQA 风格的考试题中频繁出现。本文双语详解每个主要的积分方法、关键公式及其应用。
1. Integration as Reverse Differentiation | 积分作为微分的逆运算
Integration is the inverse process of differentiation. If F'(x) = f(x), then the indefinite integral ∫ f(x) dx = F(x) + C, where C is the constant of integration. This constant accounts for the entire family of antiderivatives that differ by a constant. Understanding this inverse relationship allows you to check your integrals by differentiating the result.
积分是微分的逆过程。若 F'(x) = f(x)，则不定积分 ∫ f(x) dx = F(x) + C，其中 C 为积分常数。该常数代表所有相差一个常数的原函数族。理解这一逆运算关系后，可以通过对积分结果求导来验证答案。
2. Basic Integration Rules | 基本积分法则
Master these essential integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ aˣ dx = aˣ/ln a + C (a > 0, a ≠ 1), ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C. For linear functions, remember ∫ f(ax+b) dx often requires a factor of 1/a.
熟记以下基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)，∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C，∫ aˣ dx = aˣ/ln a + C (a > 0, a ≠ 1)，∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C，∫ sec² x dx = tan x + C。对于线性函数，牢记 ∫ f(ax+b) dx 通常需要乘以 1/a。
3. Integration by Substitution | 换元积分法
Substitution reverses the chain rule. For an integral of the form ∫ f(g(x))g'(x) dx, let u = g(x), then du = g'(x) dx, transforming the integral into ∫ f(u) du. After integrating with respect to u, substitute back to the original variable, or for definite integrals change the limits: if x goes from a to b, then u goes from g(a) to g(b). Common substitutions include u = linear expression (to simplify a fraction), u = sin x or u = cos x (with trigonometric integrals), and u = √(x). Always ensure the derivative du is fully accounted for.
换元法逆转链式法则。对于 ∫ f(g(x))g'(x) dx 形式的积分，令 u = g(x)，则 du = g'(x) dx，将积分化为 ∫ f(u) du。对 u 积分后回代原变量，或对于定积分改变积分限：若 x 从 a 到 b，则 u 从 g(a) 到 g(b)。常见换元包括 u = 线性表达式（用于简化分式）、u = sin x 或 u = cos x（三角积分），以及 u = √(x)。务必确保 du 被完整替换。
4. Integration by Parts | 分部积分法
Integration by parts stems from the product rule: ∫ u dv = uv – ∫ v du. Choose u according to the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to make ∫ v du simpler. When both functions are of the same type, strategic choice may require trial. Some integrals, such as ∫ eˣ sin x dx, need two applications and a cyclic return to the original integral, which can then be solved algebraically. Always include the constant of integration only at the final step.
分部积分源于乘法法则：∫ u dv = uv – ∫ v du。按照 LIATE 法则（对数、反三角、代数、三角、指数）选择 u，使 ∫ v du 更简单。当两个函数类型相同时，可能需要尝试选择。某些积分如 ∫ eˣ sin x dx 需两次分部并回到原积分，随后代数解出。仅需在最后一步加上积分常数。
5. Definite Integrals and the Fundamental Theorem | 定积分与基本定理
The Fundamental Theorem of Calculus links differentiation and integration: if F is any antiderivative of f on [a,b], then ∫ₐᵇ f(x) dx = F(b) – F(a). This allows exact evaluation of net signed area. The function f must be continuous on [a,b] for the theorem to apply. In practice, find an antiderivative, evaluate at the upper and lower limits, and subtract. Remember that switching limits introduces a minus sign: ∫ₐᵇ f(x) dx = –∫ᵇₐ f(x) dx.
微积分基本定理将微分与积分联系起来：若 F 是 f 在 [a,b] 上的任一原函数，则 ∫ₐᵇ f(x) dx = F(b) – F(a)。这能精确计算净有向面积。要求 f 在 [a,b] 上连续。实际计算时，找到原函数，代入上下限并相减。注意交换积分限会引入负号：∫ₐᵇ f(x) dx = –∫ᵇₐ f(x) dx。
6. Area Under a Curve | 曲线下方面积
To find the total area bounded by y = f(x) and the x-axis from x = a to x = b, use ∫ₐᵇ |f(x)| dx. Since the absolute value handles negative f(x), you must split the integral at points where f(x) = 0. Sketching the curve helps identify which parts lie above and below the axis. The signed area (integral without absolute value) yields net accumulation, which can be negative.
求 y = f(x) 与 x 轴在 x = a 到 x = b 之间围成的总面积，需使用 ∫ₐᵇ |f(x)| dx。绝对值处理了 f(x) 为负的情况，因此必须在 f(x) = 0 处拆分积分。绘制曲线草图有助于识别上方和下方区域。有向面积（不带绝对值的积分）给出净累积值，可能为负。
7. Area Between Curves | 曲线之间的面积
If two curves y = f(x) and y = g(x) satisfy f(x) ≥ g(x) on [a,b], the area between them is ∫ₐᵇ [f(x) – g(x)] dx. When the curves intersect, solve f(x) = g(x) to find the limits; the region may need to be split if the upper and lower functions swap. Always determine which is ‘top’ and ‘bottom’ by testing a point in the interval.
若两曲线 y = f(x) 与 y = g(x) 在 [a,b] 上满足 f(x) ≥ g(x)，则二者间面积为 ∫ₐᵇ [f(x) – g(x)] dx。当曲线相交时，解 f(x) = g(x) 确定积分限；若上下函数发生交换，需拆分区域。通过区间内测试点确定何为“上”和“下”。
8. Volumes of Revolution | 旋转体体积
When a region bounded by y = f(x), the x-axis, and vertical lines x = a, x = b is revolved around the x-axis, the volume is V = π ∫ₐᵇ [f(x)]² dx. For revolution around the y-axis, express x in terms of y and use V = π ∫꜀ᵈ [g(y)]² dy. If the region is bounded between two curves, use the washer method: V = π ∫ [f(x)]² – [g(x)]² dx, where f is the outer radius and g the inner radius. Always identify the axis of revolution and the limits clearly.
将由 y = f(x)、x 轴和直线 x = a、x = b 围成的区域绕 x 轴旋转，体积为 V = π ∫ₐᵇ [f(x)]² dx。绕 y 轴旋转，需将 x 表为 y 的函数，使用 V = π ∫꜀ᵈ [g(y)]² dy。若区域由两曲线围成，使用垫圈法：V = π ∫ [f(x)]² – [g(x)]² dx，其中 f 为外半径，g 为内半径。务必明确旋转轴与积分限。
9. Improper Integrals | 反常积分
Improper integrals arise when the interval of integration is infinite or the integrand has a vertical asymptote within the interval. For an infinite limit, ∫ₐ∞ f(x) dx = lim_{b→∞} ∫ₐᵇ f(x) dx. If the limit exists as a finite number, the integral converges; otherwise it diverges. For unbounded integrands, such as ∫₀¹ 1/√x dx, replace the problematic bound with a variable approaching the singularity and take the limit. Correct limit notation is essential to earn full marks.
当积分区间无穷或被积函数在区间内有垂直渐近线时，产生反常积分。对无穷限，∫ₐ∞ f(x) dx = lim_{b→∞} ∫ₐᵇ f(x) dx。若极限为有限值，积分收敛；否则发散。对于无界被积函数，如 ∫₀¹ 1/√x dx，将瑕点用趋近变量代替并取极限。正确的极限符号是获得满分的关键。
10. Kinematics Applications | 运动学应用
In rectilinear motion, given acceleration a(t) as a function of time, velocity is v(t) = ∫ a(t) dt + v₀, and displacement is s(t) = ∫ v(t) dt + s₀. Initial conditions provide the constants. The definite integral of velocity over [t₁, t₂] gives the change in displacement (displacement), while the definite integral of speed |v(t)| gives the total distance travelled. Always distinguish between these two quantities.
在直线运动中，已知加速度 a(t) 为时间函数，速度 v(t) = ∫ a(t) dt + v₀，位移 s(t) = ∫ v(t) dt + s₀。初始条件确定常数。速度在 [t₁, t₂] 上的定积分给出位移变化（位移），而速率 |v(t)| 的定积分给出总路程。务必区分这两个量。
11. Average Value of a Function | 函数的平均值
The average value of a continuous function f on the closed interval [a,b] is f_avg = 1/(b–a) ∫ₐᵇ f(x) dx. This can be interpreted as the height of a rectangle with the same area as the region under the curve. Typical exam questions ask you to find the average value and then solve for c in [a,b] such that f(c) = f_avg—a direct application of the Mean Value Theorem for Integrals.
连续函数 f 在闭区间 [a,b] 上的平均值为 f_avg = 1/(b–a) ∫ₐᵇ f(x) dx。这可解释为与曲线下方面积相等的矩形的高度。典型考题要求计算平均值并求解 c ∈ [a,b] 使 f(c) = f_avg，直接运用积分中值定理。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always add the constant +C in indefinite integrals. In substitution, check that du is correctly derived and that no factor is lost. For area problems, break the integral where the function crosses the x-axis to avoid negative cancellation. In volumes of revolution, do not forget the factor π and square the function correctly. When using the fundamental theorem, verify continuity. Practice with past AQA-style questions to spot patterns like repeated integration by parts or clever substitutions. Show clear steps and use proper mathematical notation, especially for limits in improper integrals.
不定积分务必添加常数 +C。换元时核查 du 是否正确求导，无因子遗漏。面积问题中在函数穿过 x 轴处拆分积分，避免负面积抵消。旋转体体积切勿遗漏 π，并正确平方函数。使用微积分基本定理时验证连续性。多练习 AQA 风格历年真题，识别重复分部积分或巧妙换元等模式。展示清晰步骤，并使用正确的数学符号，尤其是反常积分中的极限表达。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导